cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 13 results. Next

A163280 Square array read by antidiagonals where column k lists the numbers j whose largest divisor <= sqrt(j) is k.

Original entry on oeis.org

1, 2, 4, 3, 6, 9, 5, 8, 12, 16, 7, 10, 15, 20, 25, 11, 14, 18, 24, 30, 36, 13, 22, 21, 28, 35, 42, 49, 17, 26, 27, 32, 40, 48, 56, 64, 19, 34, 33, 44, 45, 54, 63, 72, 81, 23, 38, 39, 52, 50, 60, 70, 80, 90, 100, 29, 46, 51, 68, 55, 66, 77, 88, 99, 110, 121, 31, 58, 57, 76, 65, 78, 84, 96, 108, 120, 132, 144
Offset: 1

Views

Author

Omar E. Pol, Aug 07 2009

Keywords

Comments

This sequence is a permutation of the natural numbers A000027. Note that the first column is formed by 1 together with the prime numbers.
Column k contains exactly those numbers j=k*m where m is either a prime >= j or one of the numbers in row k of A163925. - Franklin T. Adams-Watters, Aug 12 2009

Examples

			Array begins:
   1,  4,  9,  16,  25,  36,  49,  64,  81, 100, 121, 144, ...
   2,  6, 12,  20,  30,  42,  56,  72,  90, 110, 132, 156, ...
   3,  8, 15,  24,  35,  48,  63,  80,  99, 120, 143, 168, ...
   5, 10, 18,  28,  40,  54,  70,  88, 108, 130, 154, 180, ...
   7, 14, 21,  32,  45,  60,  77,  96, 117, 140, 165, 192, ...
  11, 22, 27,  44,  50,  66,  84, 104, 126, 150, 176, 204, ...
  13, 26, 33,  52,  55,  78,  91, 112, 135, 160, 187, 216, ...
  17, 34, 39,  68,  65, 102,  98, 128, 153, 170, 198, 228, ...
  19, 38, 51,  76,  75, 114, 105, 136, 162, 190, 209, 264, ...
  23, 46, 57,  92,  85, 138, 119, 152, 171, 200, 220, 276, ...
  29, 58, 69, 116,  95, 174, 133, 184, 189, 230, 231, 348, ...
  31, 62, 87, 124, 115, 186, 147, 232, 207, 250, 242, 372, ...
  ...

Links

Crossrefs

Rows 1-12: A000290, A002378, A005563, A164004, A100451, A164006, A164007, A164008, A164009, A164010, A164011, A164012.
Columns 1-12: A008578, A161344, A161345, A161424, A161835, A162526, A162527, A162528, A162529, A162530, A162531, A162532.
Another version: A163990.
Cf. A000027, A000040, A033676, A147861, A163100, A164000.

Programs

  • Maple
    A163280 := proc(n,k) local r,T ; r := 0 ; for T from k^2 by k do if A033676(T) = k then r := r+1 ; if r = n then RETURN(T) ; fi; fi; od: end: # R. J. Mathar, Aug 09 2009
  • Mathematica
    nmax = 12;
pm = Prime[nmax];
sDiv[n_] := Select[Divisors[n], #^2 <= n&][[-1]];
Clear[col]; col[k_] := col[k] = Select[Range[k pm], sDiv[#] == k&];
T[n_, k_ /; 1 <= k <= Length[col[k]]] := col[k][[n]];
Table[T[n-k+1, k], {n, 1, nmax}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 15 2019 *)

Formula

Column k lists the numbers j such that A033676(j)=k.

Extensions

Edited by R. J. Mathar, Aug 01 2010
Example edited by Jean-François Alcover, Dec 15 2019

A163990 Square array read by antidiagonals where the row n lists the numbers k such that their largest divisor <= sqrt(k) equals n.

Original entry on oeis.org

1, 4, 2, 9, 6, 3, 16, 12, 8, 5, 25, 20, 15, 10, 7, 36, 30, 24, 18, 14, 11, 49, 42, 35, 28, 21, 22, 13, 64, 56, 48, 40, 32, 27, 26, 17, 81, 72, 63, 54, 45, 44, 33, 34, 19, 100, 90, 80, 70, 60, 50, 52, 39, 38, 23, 121, 110, 99, 88, 77, 66, 55, 68, 51, 46, 29, 144, 132, 120, 108
Offset: 1

Views

Author

Omar E. Pol, Aug 11 2009

Keywords

Comments

This sequence is a permutation of the natural numbers.
Note that the first row is formed by 1 together the prime numbers and the first column are the squares of the natural numbers.
For more information see A163280, the main entry for this sequence. (See also A161344).

Examples

			Array begins:
1, 2, 3, 5, 7, 11,
4, 6, 8, 10, 14,
9, 12, 15, 18,
16, 20, 24,
25, 30,
36,
See also the array in A163280.

Links

Crossrefs

Cf. A000027, A000040, A033676, A147861, A163100, A163280, A163281, A163991, A164000.
Cf. Rows: 1=A008578, 2=A161344, 3=A161345, 4=A161424, 5=A161835, 6=A162526, 7=A162527, 8=A162528, 9=A162529, 10=A162530, 11=A162531, 12=A162532.
Cf. Columns: 1=A000290, 2=A002378, 4=A164004, 6=A164006, 7=A164007, 8=A164008, 9=A164009, 10=A164010, 11=A164011, 12=A164012.

Formula

Row n lists the numbers k such that A033676(k)=n.

A164004 Zero together with row 4 of the array in A163280.

Original entry on oeis.org

0, 5, 10, 18, 28, 40, 54, 70, 88, 108, 130, 154, 180, 208, 238, 270, 304, 340, 378, 418, 460, 504, 550, 598, 648, 700, 754, 810, 868, 928, 990, 1054, 1120, 1188, 1258, 1330, 1404, 1480, 1558, 1638, 1720, 1804, 1890, 1978, 2068, 2160, 2254, 2350, 2448, 2548
Offset: 0

Views

Author

Omar E. Pol, Aug 08 2009

Keywords

Links

Crossrefs

Cf. A008578, A161344, A161345, A163280, A164000, A005563, A164005.
Cf. A028552.

Programs

  • Maple
    A033676 := proc(n) local a,d; a := 0 ; for d in numtheory[divisors](n) do if d^2 <= n then a := max(a,d) ; fi; od: a; end: A163280 := proc(n,k) local r,T ; r := 0 ; for T from k^2 by k do if A033676(T) = k then r := r+1 ; if r = n then RETURN(T) ; fi; fi; od: end: A164004 := proc(n) if n = 0 then 0; else A163280(4,n) ; fi; end: seq(A164004(n),n=0..80) ; # R. J. Mathar, Aug 09 2009
  • Mathematica
    Join[{0, 5}, Table[n*(n + 3), {n, 2, 50}]] (* G. C. Greubel, Aug 28 2017 *)
  • PARI
    x='x+O('x^50); concat([0], Vec(x*(x^3 -3*x^2 +5*x -5)/(x-1)^3)) \\ G. C. Greubel, Aug 28 2017

Formula

Conjectures from Colin Barker, Apr 07 2015: (Start)
a(n) = n*(3+n) = A028552(n) for n > 1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 4.
G.f.: x*(x^3 - 3*x^2 + 5*x - 5) / (x-1)^3. (End)
E.g.f.: x*(x+4)*exp(x) + x. - G. C. Greubel, Aug 28 2017

Extensions

Extended beyond a(12) by R. J. Mathar, Aug 09 2009

A164006 Zero together with row 6 of the array in A163280.

Original entry on oeis.org

0, 11, 22, 27, 44, 50, 66, 84, 104, 126, 150, 176, 204, 234, 266, 300, 336, 374, 414, 456, 500, 546, 594, 644, 696, 750, 806, 864, 924, 986, 1050, 1116, 1184, 1254, 1326, 1400, 1476, 1554, 1634, 1716, 1800, 1886, 1974, 2064, 2156, 2250, 2346, 2444, 2544, 2646
Offset: 0

Views

Author

Omar E. Pol, Aug 08 2009

Keywords

Links

Crossrefs

Cf. A008578, A161344, A161345, A163280, A164000, A164005, A164007.
Cf. A028557 for n > 4. - R. J. Mathar, Aug 09 2009

Programs

  • Maple
    A033676 := proc(n) local a,d; a := 0 ; for d in numtheory[divisors](n) do if d^2 <= n then a := max(a,d) ; fi; od: a; end: A163280 := proc(n,k) local r,T ; r := 0 ; for T from k^2 by k do if A033676(T) = k then r := r+1 ; if r = n then RETURN(T) ; fi; fi; od: end: A164006 := proc(n) if n = 0 then 0; else A163280(6,n) ; fi; end: seq(A164006(n),n=0..80) ; # R. J. Mathar, Aug 09 2009
  • Mathematica
    Join[{0,11,22,27}, Table[n*(n + 5), {n, 4, 50}]] (* G. C. Greubel, Aug 28 2017 *)
  • PARI
    concat(0, Vec(x*(8*x^6-21*x^5+23*x^4-18*x^3+6*x^2+11*x-11)/(x-1)^3 + O(x^100))) \\ Colin Barker, Nov 24 2014

Formula

From Colin Barker, Nov 24 2014: (Start)
a(n) = n*(n+5) for n > 4.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 7.
G.f.: x*(8*x^6 - 21*x^5 + 23*x^4 - 18*x^3 + 6*x^2 + 11*x - 11) / (x-1)^3. (End)
E.g.f.: (x/2)*(10 + 8*x + x^2 + 2*(x + 6)*exp(x)). - G. C. Greubel, Aug 28 2017

Extensions

Extended beyond a(12) by R. J. Mathar, Aug 09 2009

A164007 Zero together with row 7 of the array in A163280.

Original entry on oeis.org

0, 13, 26, 33, 52, 55, 78, 91, 112, 135, 160, 187, 216, 247, 280, 315, 352, 391, 432, 475, 520, 567, 616, 667, 720, 775, 832, 891, 952, 1015, 1080, 1147, 1216, 1287, 1360, 1435, 1512, 1591, 1672, 1755, 1840, 1927, 2016, 2107, 2200, 2295, 2392, 2491, 2592, 2695
Offset: 0

Views

Author

Omar E. Pol, Aug 08 2009

Keywords

Links

Crossrefs

Cf. A008578, A161344, A161345, A163280, A164000, A164006, A164008.
Cf. A028560 for n > 6.

Programs

  • Maple
    A033676 := proc(n) local a,d; a := 0 ; for d in numtheory[divisors](n) do if d^2 <= n then a := max(a,d) ; fi; od: a; end: A163280 := proc(n,k) local r,T ; r := 0 ; for T from k^2 by k do if A033676(T) = k then r := r+1 ; if r = n then RETURN(T) ; fi; fi; od: end: A164007 := proc(n) if n = 0 then 0; else A163280(7,n) ; fi; end: seq(A164007(n),n=0..80) ;  # R. J. Mathar, Aug 09 2009
  • Mathematica
    Join[{0, 13, 26, 33, 52, 55, 78}, Table[n*(n + 6), {n, 7, 50}]] (* G. C. Greubel, Aug 28 2017 *)
LinearRecurrence[{3,-3,1},{0,13,26,33,52,55,78,91,112,135},50] (* Harvey P. Dale, Jul 03 2020 *)
  • PARI
    my(x='x+O('x^50)); concat([0], Vec(x*(13 - 13*x - 6*x^2 + 18*x^3 - 28*x^4 + 36*x^5 - 30*x^6 + 18*x^7 - 6*x^8)/(1 - x)^3)) \\ G. C. Greubel, Aug 28 2017

Formula

From G. C. Greubel, Aug 28 2017: (Start)
a(n) = n*(n+6), n >= 7.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), n >= 7.
G.f.: x*(13 - 13*x - 6*x^2 + 18*x^3 - 28*x^4 + 36*x^5 - 30*x^6 + 18*x^7 - 6*x^8)/(1 - x)^3.
E.g.f.: (7*x + x^2)*exp(x) + 6*x +5*x^2 + x^3 + x^4/2 + x^6/120. (End)

Extensions

Extended by R. J. Mathar, Aug 09 2009

A164008 Zero together with row 8 of the array in A163280.

Original entry on oeis.org

0, 17, 34, 39, 68, 65, 102, 98, 128, 153, 170, 198, 228, 260, 294, 330, 368, 408, 450, 494, 540, 588, 638, 690, 744, 800, 858, 918, 980, 1044, 1110, 1178, 1248, 1320, 1394, 1470, 1548, 1628, 1710, 1794, 1880, 1968, 2058, 2150, 2244, 2340, 2438, 2538, 2640
Offset: 0

Views

Author

Omar E. Pol, Aug 08 2009

Keywords

Links

Crossrefs

Cf. A008578, A161344, A161345, A163280, A164000, A164007, A164009.

Programs

  • Maple
    A033676 := proc(n) local dvs; dvs := sort(convert(numtheory[divisors](n), list)) ; op(floor((nops(dvs)+1)/2) , dvs) ; end: A163280 := proc(n, k) local r, T ; r := 0 ; for T from k^2 by k do if A033676(T) = k then r := r+1 ; if r = n then RETURN(T) ; fi; fi; od: end: printf("0,") ; for n from 1 to 70 do printf("%d,",A163280(8,n)) ; end do ; # R. J. Mathar, Feb 05 2010
  • Mathematica
    LinearRecurrence[{3,-3,1},{0,17,34,39,68,65,102,98,128,153,170,198,228},50] (* Harvey P. Dale, Dec 25 2022 *)

Formula

Conjecture: a(n) = A028563(n), n > 9. [R. J. Mathar, Jul 31 2010]

Extensions

Terms beyond 228 from R. J. Mathar, Feb 05 2010

A164009 Zero together with row 9 of the array in A163280.

Original entry on oeis.org

0, 19, 38, 51, 76, 75, 114, 105, 136, 162, 190, 209, 264, 273, 308, 345, 384, 425, 468, 513, 560, 609, 660, 713, 768, 825, 884, 945, 1008, 1073, 1140, 1209, 1280, 1353, 1428, 1505, 1584, 1665, 1748, 1833, 1920, 2009, 2100, 2193, 2288, 2385, 2484, 2585, 2688
Offset: 0

Views

Author

Omar E. Pol, Aug 08 2009

Keywords

Links

Crossrefs

Cf. A008578, A161344, A161345, A163280, A164000, A164008, A164010.

Programs

  • Maple
    A033676 := proc(n) local dvs; dvs := sort(convert(numtheory[divisors](n), list)) ; op(floor((nops(dvs)+1)/2) , dvs) ; end proc:
A163280 := proc(n, k) local r, T ; r := 0 ; for T from k^2 by k do if A033676(T) = k then r := r+1 ; if r = n then return T ; end if; end if; end do: end proc:
printf("0, ") ; for n from 1 to 90 do printf("%d, ", A163280(9, n)) ; end do ; # R. J. Mathar, Jul 31 2010

Formula

Conjecture: a(n) = A028566(n), n > 12. [R. J. Mathar, Jul 31 2010]

Extensions

Terms beyond a(12) from R. J. Mathar, Jul 31 2010

A164010 Zero together with row 10 of the array in A163280.

Original entry on oeis.org

0, 23, 46, 57, 92, 85, 138, 119, 152, 171, 200, 220, 276, 286, 322, 375, 416, 442, 486, 532, 580, 630, 682, 736, 792, 850, 910, 972, 1036, 1102, 1170, 1240, 1312, 1386, 1462, 1540, 1620, 1702, 1786, 1872, 1960, 2050, 2142, 2236, 2332, 2430, 2530, 2632, 2736
Offset: 0

Views

Author

Omar E. Pol, Aug 08 2009

Keywords

Links

Crossrefs

Cf. A008578, A161344, A161345, A163280, A164000, A164009, A164011.

Formula

Conjecture: a(n) = A028569(n), n > 16. [R. J. Mathar, Jul 31 2010]

Extensions

Terms beyond a(12) from R. J. Mathar, Feb 06 2010

A164011 Zero together with row 11 of the array in A163280.

Original entry on oeis.org

0, 29, 58, 69, 116, 95, 174, 133, 184, 189, 230, 231, 348, 299, 350, 390, 448, 459, 522, 551, 620, 651, 704, 759, 816, 875, 936, 999, 1064, 1131, 1200, 1271, 1344, 1419, 1496, 1575, 1656, 1739, 1824, 1911, 2000, 2091, 2184, 2279, 2376, 2475, 2576, 2679, 2784
Offset: 0

Views

Author

Omar E. Pol, Aug 08 2009

Keywords

Links

Crossrefs

Cf. A008578, A161344, A161345, A163280, A164000, A164010, A164012.

Programs

  • Maple
    A033676 := proc(n) local dvs; dvs := sort(convert(numtheory[divisors](n), list)) ; op(floor((nops(dvs)+1)/2) , dvs) ; end: A163280 := proc(n, k) local r, T ; r := 0 ; for T from k^2 by k do if A033676(T) = k then r := r+1 ; if r = n then RETURN(T) ; fi; fi; od: end: printf("0,") ; for n from 1 to 70 do printf("%d,",A163280(11,n)) ; end do ; # R. J. Mathar, Feb 05 2010

Formula

Conjecture: a(n) = A098603(n), n > 20. [R. J. Mathar, Jul 31 2010]

Extensions

Extended by R. J. Mathar, Feb 05 2010

A164005 Zero together with row 5 of the array in A163280.

Original entry on oeis.org

0, 7, 14, 21, 32, 45, 60, 77, 96, 117, 140, 165, 192, 221, 252, 285, 320, 357, 396, 437, 480, 525, 572, 621, 672, 725, 780, 837, 896, 957, 1020, 1085, 1152, 1221, 1292, 1365, 1440, 1517, 1596, 1677, 1760, 1845, 1932, 2021, 2112, 2205, 2300, 2397, 2496, 2597
Offset: 0

Views

Author

Omar E. Pol, Aug 08 2009

Keywords

Links

Crossrefs

Cf. A008578, A161344, A161345, A163280, A164000, A164004, A164006.
Cf. A028347.

Programs

  • Maple
    A033676 := proc(n) local a,d; a := 0 ; for d in numtheory[divisors](n) do if d^2 <= n then a := max(a,d) ; fi; od: a; end: A163280 := proc(n,k) local r,T ; r := 0 ; for T from k^2 by k do if A033676(T) = k then r := r+1 ; if r = n then RETURN(T) ; fi; fi; od: end: A164005 := proc(n) if n = 0 then 0; else A163280(5,n) ; fi; end: seq(A164005(n),n=0..80) ; # R. J. Mathar, Aug 09 2009
  • Mathematica
    Join[{0, 7, 14}, Table[n*(n + 4), {n, 3, 50}]] (* G. C. Greubel, Aug 28 2017 *)
  • PARI
    x='x+O('x^50); concat([0], Vec(x*(7 - 7*x + 4*x^3 - 2*x^4)/(1 - x)^3)) \\ G. C. Greubel, Aug 28 2017

Formula

Conjecture: a(n) = A100451(n+2). (See A163280.)
From G. C. Greubel, Aug 28 2017: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), n >= 3.
a(n) = n*(n+4), n >= 3.
G.f.: x*(7 - 7*x + 4*x^3 - 2*x^4)/(1 - x)^3.
E.g.f.: x*(x+5)*exp(x) + 2*x + x^2. (End)

Extensions

Extended by R. J. Mathar, Aug 09 2009
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